The homotopy type of hyperplane posets

Edelman, Paul H.; Walker, James W.

doi:10.1090/S0002-9939-1985-0784167-3

The homotopy type of hyperplane posets
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by Paul H. Edelman and James W. Walker

Proc. Amer. Math. Soc. 94 (1985), 221-225

DOI: https://doi.org/10.1090/S0002-9939-1985-0784167-3

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Abstract:

Previously, Edelman had defined a partial order on the regions of a euclidean space dissected by hyperplanes. The goal of this paper is to compute the homotopy type of open intervals in these posets. Techniques from the theory of shellable posets are used.

References

Anders Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), no. 1, 159–183. MR 570784, DOI 10.1090/S0002-9947-1980-0570784-2
Anders Björner, Orderings of Coxeter groups, Combinatorics and algebra (Boulder, Colo., 1983) Contemp. Math., vol. 34, Amer. Math. Soc., Providence, RI, 1984, pp. 175–195. MR 777701, DOI 10.1090/conm/034/777701
H. Bruggesser and P. Mani, Shellable decompositions of cells and spheres, Math. Scand. 29 (1971), 197–205 (1972). MR 328944, DOI 10.7146/math.scand.a-11045
Anders Björner and Michelle Wachs, On lexicographically shellable posets, Trans. Amer. Math. Soc. 277 (1983), no. 1, 323–341. MR 690055, DOI 10.1090/S0002-9947-1983-0690055-6
Raul Cordovil, A combinatorial perspective on the non-Radon partitions, J. Combin. Theory Ser. A 38 (1985), no. 1, 38–47. MR 773553, DOI 10.1016/0097-3165(85)90019-6
Paul H. Edelman, A partial order on the regions of $\textbf {R}^{n}$ dissected by hyperplanes, Trans. Amer. Math. Soc. 283 (1984), no. 2, 617–631. MR 737888, DOI 10.1090/S0002-9947-1984-0737888-6

Decompositions, shellings, and diameters of simplicial complexes and convex polyhedra

Daniel Quillen, Homotopy properties of the poset of nontrivial $p$-subgroups of a group, Adv. in Math. 28 (1978), no. 2, 101–128. MR 493916, DOI 10.1016/0001-8708(78)90058-0
James W. Walker, Homotopy type and Euler characteristic of partially ordered sets, European J. Combin. 2 (1981), no. 4, 373–384. MR 638413, DOI 10.1016/S0195-6698(81)80045-5